Methods exist for demodulating fringe patterns. Examples of such patterns are illustrated in FIGS. 5 and 14A. However, conventional methods of demodulating closed curve fringe patterns, including circular and elliptical fringe patterns, inevitably produce ambiguities. Ambiguities are then typically resolved by including additional constraints. These additional constraints may include restrictions on the smoothness of curves within the fringe patterns.
In addition to ambiguities other defects and errors may also be produced by the preliminary demodulation process. An example of this can be found in the classic Fourier (Hilbert) transform method (FTM). When the FTM is applied to a whole image, artefacts are produced, the artefacts being related to discontinuities introduced by the half-plane filter used in Fourier space. The artefacts result in errors in phase estimation, and specifically in regions where an angle of the fringe pattern is close to perpendicular with the half-plane discontinuity line.
A method adopted to overcome this problem is to choose the half-plane filter in Fourier space such that the discontinuity line does not cut through the signal in Fourier space.
However, the frequency components derived from circular or elliptical fringe patterns are also “circular” and no discontinuity line could be defined which does not cut through the frequency signal.
Other methods based on local (small kernel) demodulation may avoid the errors of the FTM, but typically fail to disambiguate correctly when there are strong perturbations in the underlying fringe pattern. Features such as bifurcations and fringe endings represent typically strong perturbations.
Images are often characterised by a number of features including pattern and texture. A number of image processing operations depend upon the estimation of the orientation of these image features.
Image processing/analysis applications where the estimation of orientation is important include:                edge orientation detection;        orientation input to steerable filters;        pattern analysis;        texture analysis, where the local orientation of texture is estimated for characterisation purposes; and        orientation selective filtering, processing, and enhancement of images.        
One such area where the estimation of orientation is particularly important is the demodulation of fringe patterns or equivalently, AM-FM modulated patterns. One area where such patterns exist is in fingerprint analysis. Other areas include the analysis of naturally formed patterns such as an iris diaphragm, ripples in sand, and fluid convection in thin layers. The primary reason why it is useful to know the fringe orientation when demodulating such a fringe pattern is because it allows for 1-dimensional demodulators to be aligned correctly.
Another area where the estimation of pattern orientation is used is in image re-sampling and enhancement.
Known methods for estimating pattern orientation include gradient-based methods. A serious deficiency with such gradient-based methods is that at the ridges and valleys of images, the gradient has zero magnitude. Gradient methods are therefore unable to provide information about pattern direction in such regions.
Interferometry refers to experiments involving the interference of two light waves that have suffered refraction, diffraction or reflection in the interferometer. Such experiments typically involve the testing of lenses, mirrors and other optical components. The principle parameter of interest in interferometry is the phase difference between interfering beams.
Present methods of demodulating sequences of phase-related fringe-patterns generally rely on a priori knowledge of the phase-shifts between each of the individual patterns. When these relative phase-shifts are available, it is possible to estimate the spatial phase by using a generalised phase-shifting algorithm (PSA). In cases where the relative phase-shifts are not a priori known, the estimated spatial phase will be incorrect unless special error-reducing PSAs are used or methods to estimate the actual phase-shift between patterns are used.
Error-reducing PSAs are useful where the deviation of the actual phase-shift from the expected phase-shift is small (typically less than 0.1 radian) and deterministic. When the phase-shift deviation is larger and/or non-deterministic (ie essentially random), then other methods must be used.
Methods have been developed to estimate phase shifts between phase-related fringe patterns. Certain of the methods, based on statistical (least squares) estimation, requires at least 5 fringe patterns to work. Methods based on fitting the phase shifts to an ellipse also require at least 5 fringe patterns. Methods based on the statistical properties of spatial phase histograms have been proposed, but are heavily dependent on certain restrictive, and often unrealistic criteria. More recently methods based upon the correlation between patterns have been proposed but are very sensitive to the number of full fringes in the frames. These correlation methods assume that the fringes are essentially linear in the frame, which means only trivial fringe patterns can be processed. Methods based on the maximum and minimum variation of individual pixels in different patterns typically require at least 15 frames to operate effectively.
Accordingly, current methods of demodulating sequences of phase-related fringe-patterns are unsatisfactory in many situations, especially for a small number of phase-related fringe-patterns, and many algorithms do not work in the case of three or four frames.